Stability conditions and $\mu$-stable sheaves on K3 surfaces with Picard number one

TL;DR

This paper explores the stability of certain sheaves on K3 surfaces with Picard number one, demonstrating that some semi-rigid sheaves are Bridgeland stable and that the stability space can reconstruct the surface.

Contribution

It shows that some semi-rigid μ-stable sheaves are Bridgeland stable on K3 surfaces with Picard number one, and that the stability space encodes the surface itself.

Findings

Semi-rigid μ-stable sheaves are Bridgeland stable.

The stability space U(X) reconstructs the K3 surface.

Contrasts with the abelian surface case.

Abstract

In this article, we show that some semi-rigid $\mu$-stable sheaves on a projective K3 surface $X$ with Picard number 1 are stable in the sense of Bridgeland's stability condition. As a consequence of our work, we show that the special set $U(X) \subset \Stab (X)$ reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.